Question

$f(z)= \frac{z}{1-e^{-z}}$ $\Leftrightarrow$ $\forall n \ge 0$ and $n\in\mathbb{Z}$, coefficient of $z^n$ in $f^{n+1}(z)$ is $1$.

score 196 · Answer 1 · 2026-04-16 13:00:16

196

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$f(z)= \frac{z}{1-e^{-z}}$ $\Leftrightarrow$ $\forall n \ge 0$ and $n\in\mathbb{Z}$, coefficient of $z^n$ in $f^{n+1}(z)$ is $1$.

Published on 16 Apr 2026 - 13:00

score 1.7k · Answer 2 · 2026-04-11 21:56:59

1.7k

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Complex solutions of $e^{e^z}=1$

Published on 11 Apr 2026 - 21:56

score 1.8k · Answer 3 · 2026-04-13 00:47:00

1.8k

Views

Proving quotient rule in the complex plane

Published on 13 Apr 2026 - 0:47

score 2.2k · Answer 4 · 2026-04-11 17:28:12

2.2k

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Find the principal value of $\big[\frac{e}{2}(-1-\sqrt3i)\big]^{3\pi i}$

Published on 11 Apr 2026 - 17:28

score 206 · Answer 5 · 2026-04-15 15:00:20

206

Views

Classify singularities of $f(z) = \frac{z^5}{1+z+z^2+z^3+z^4}$

Published on 15 Apr 2026 - 15:00

score 1k · Answer 6 · 2026-04-15 16:49:17

1k

Views

Let $P(z)=1+2z+3z^2+\cdots+nz^{n-1}$. Use the Gauss-Lucas theorem to show that all roots of this polynomial are in the unit disk.

Published on 15 Apr 2026 - 16:49

score 95 · Answer 7 · 2026-03-25 07:38:22

95

Views

Evaluate infinite integral using complex analysis

Published on 25 Mar 2026 - 7:38

score 563 · Answer 8 · 2026-04-11 13:56:33

563

Views

Partial fraction decomposition for complex polynomials

Published on 11 Apr 2026 - 13:56

score 1.4k · Answer 9 · 2026-04-13 17:24:21

1.4k

Views

Prove that $\sin(z) =\frac{e^{iz}-e^{-iz}}{2i} =\sin(x)\cosh(y) +i\cos(x)\sinh(y)$

Published on 13 Apr 2026 - 17:24

score 425 · Answer 10 · 2018-04-10 04:51:44

425

Views

Check that $f(0)=\frac{1}{2}$ using Schwarz Lemma

Published on 10 Apr 2018 - 4:51

score 433 · Answer 11 · 2026-04-11 12:59:34

433

Views

Simply connected holomorphic functions

Published on 11 Apr 2026 - 12:59

score 779 · Answer 12 · 2026-04-11 12:34:21

779

Views

How to show that a function is meromorphic?

Published on 11 Apr 2026 - 12:34

score 177 · Answer 13 · 2026-04-17 00:41:40

177

Views

Find the coefficient of $x^{26}$ in $(x^2+x^3+x^4+x^5+x^6)^8$.

Published on 17 Apr 2026 - 0:41

score 45 · Answer 14 · 2026-04-15 22:06:18

45

Views

Mobius transformation that maps upper part of plane to lower part of plane

Published on 15 Apr 2026 - 22:06

score 539 · Answer 15 · 2026-04-12 05:41:48

539

Views

Fourier series convergence in $C^\infty(\mathbb{R}^N)$

Published on 12 Apr 2026 - 5:41

List Question

$f(z)= \frac{z}{1-e^{-z}}$ $\Leftrightarrow$ $\forall n \ge 0$ and $n\in\mathbb{Z}$, coefficient of $z^n$ in $f^{n+1}(z)$ is $1$.

Complex solutions of $e^{e^z}=1$

Proving quotient rule in the complex plane

Find the principal value of $\big[\frac{e}{2}(-1-\sqrt3i)\big]^{3\pi i}$

Classify singularities of $f(z) = \frac{z^5}{1+z+z^2+z^3+z^4}$

Let $P(z)=1+2z+3z^2+\cdots+nz^{n-1}$. Use the Gauss-Lucas theorem to show that all roots of this polynomial are in the unit disk.

Evaluate infinite integral using complex analysis

Partial fraction decomposition for complex polynomials

Prove that $\sin(z) =\frac{e^{iz}-e^{-iz}}{2i} =\sin(x)\cosh(y) +i\cos(x)\sinh(y)$

Check that $f(0)=\frac{1}{2}$ using Schwarz Lemma

Simply connected holomorphic functions

How to show that a function is meromorphic?

Find the coefficient of $x^{26}$ in $(x^2+x^3+x^4+x^5+x^6)^8$.

Mobius transformation that maps upper part of plane to lower part of plane

Fourier series convergence in $C^\infty(\mathbb{R}^N)$

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